Tutorial
Session Six – Three dimensional Semi-variograms
The
example session with EcoSSe which is described below is intended as
an example run to familiarise the user with the package. This documented
example illustrates one possible set of analyses which may be carried out. One
of the most neglected aspects of statistical analysis --- especially of spatial
data --- is the purely visual assessment of the sample data. It takes you
through the following sequence of analyses:
Ø Calculating
and interpreting a semi-variogram
There
are many other facilities within the package, which are given as alternative
options on the menus. To start the tutorial, choose EcoSSe from your Start menu. See
Tutorial One for starting up and specifying your ghost file output.
As
you can see from the above I have elected to read in a set of sample data by
clicking on the 
option and selecting 
from the menu which appears. EcoSSe
will remember the last five data files accessed and include these in your
options. Three input file types can be read in. I will read in a standard
Geostokos data file.
The
layout of such files is described in detail in the main EcoSSe documentation. The routine
which reads in the data shows the first 10 lines of your data file so that you
can check it is going in OK. The routine also checks whether we actually had
the correct number of samples on the file and informs you if there is any discrepancy.
Even
if you select a file from the list of previously analysed data files, EcoSSe
will ask you to confirm your choice. This is actually a quick way of getting
back to your working directory, since you can change your choice at this point.
Be warned, though, that if you change which file you want to read it must be
the same type of file – that is, if you are reading a standard Geostokos data
file, you cannot change your mind at this point and read in a CSV type file.
For
this illustration, I have selected copper.dat for my input data file.
This is a set of 442 borehole samples drilled into an unspecified mine dump in
As
your data is read in, it is stored on a working binary file. A progress bar
will indicate how far the process has gone. When data input is complete, your
Window should look like this:
Note
that this data set also includes ‘geology’ or ‘zone’ coding. A column on the
data file includes an integer code which was of some meaning to the original
logger. Perhaps a lithology or a dating code (since this is a dump). This
enables you to select data by code for each analysis and has the effect of
‘separating’ samples in analyses such as semi-variogram calculation.
Semi-variogram calculation
When
the data has been read in, you will see that the “greyed out” options on the
main menu bar will be activated. We use the menu bar to select an option,
say:
The
screen will prompt you to choose the four variables for the analysis – X, Y and
Z co-ordinates and the measurement which is to be analysed.
The
routine, needs to have information on the position of the samples and on the
value at each sample location. This particular data file only contains three
variables. However, at this stage, EcoSSe does not know which of these variables is
which.
You
will see two dialog boxes. The one in the top left hand corner lists the
variables available for analysis in your data file and the bottom right box
shows the variables already chosen (at this point, none!).
There
is a lot of information on the screen. At the bottom of the Window, you see the
“status bar” which shows the name of the current data file and the title read
from that file. The “already chosen” dialog box shows you that you are expected
to select variables to be the “X (east/west) co-ordinate”, “Y (north/south)
co-ordinate” and “Measurement to be analysed” for your semi-variogram.
The
upper left dialog box lists the variable names as they appeared in the data
file and is prompting you to choose the variable which will be the “X
co-ordinate” on the graph. For this example, let us choose X co-ordinate for
the X co-ordinate:
We
may then choose “Y co-ordinate” for the Y co-ordinate:
Since
we are working in three dimensions, we need a third co-ordinate:
We
must choose the variable to be analysed and state any relevant transformations
to be made. For this data we require no transformation of the variable “Grade
(%Cu)”, so click on 
.
|
|
|
Since
this data file includes ‘zone’ codes, you will need to specify which codes are
to be included in the analysis and whether the routine should distinguish
between them.
You
may select any combination of zone codes by clicking in the check boxes.
Once
selected, use the 
to tell the routine whether or not to
distinguish between samples with different zone codes. The semi-variogram
calculation will not include pairs of samples with different codes unless you
check this box.
For
this illustration, we will select all codes by clicking on 
and check the ‘ignore code’ box:
Once
we click on 
,
the 
dialog will
show the complete set of chosen variables. You still have the option to change
your mind here by clicking on 
.
This
choice of variables is acceptable, so click on 
to proceed. This may seem tedious to you at
the moment, but (later) try running the program with another set of data with
more variables. Or try a data set where the columns are in a different order.
The EcoSSe
input routine has been written to allow you this flexibility in building
your data files.
Now,
we may finally proceed to calculating a semi-variogram. For the complete data
set, samples are paired up. The difference between the values of the two
samples is calculated and squared. Plotting each of these points on a graph – squared difference versus
distance – results in a “variogram cloud”.
For
the semi-variogram interpretation and modelling routines the “differences” are
grouped together into “distance” intervals. That is, all pairs of samples which
are more or less the same distance apart are grouped together and the
differences averaged. To do this, you must choose a distance interval and a
number of groups. The maximum distance considered will be the product of these
two values.
You
have the opportunity to specify your own directions, in which case you will
need to define direction as azimuth
clockwise from North and dip down
from horizontal. For user defined directions, you must also specify a
tolerance angle to be allowed on either side of your specified azimuth and dip
directions.
You
can simply make a graph which ignores direction entirely and groups all possible
pairs of samples into one semi-variogram.
Alternatively,
you may accept the thirteen default directions: 0", 45", 90" and 135" (North, Northeast, East and
Southeast) horizontal, 0", 45", 90", 135", 180", 225", 270" and 315" at a dip of 45" down from horizontal plus a
‘vertical’ semi-variogram for all directions dip 90". If you choose the “main
points of the compass” you will also get the “omni-directional”
semi-variogram. The default directions
allow 22.5" either side, so that the
four directions cover all possibilities.
For
borehole data, accepting the default directions may not be appropriate, since
you have to select a single interval width to be used in all directions. For this illustration, we will choose our own
directions and specify azimuth, dip and tolerances as well as interval width
and number of intervals.
It
is a good idea to have run an exercise such as that described in Tutorial Two,
to have a good idea of your inherent sample spacing and the extent of your
study area before selecting interval widths and maximum distances.
Click
on 
and a new dialog will appear.
You
can define up to 24 different directions by scrolling along the bottom of the
dialog. I will specify two directions only for this example:
1. all possible pairs in the
horizontal directions: azimuth 90" ± 90", dip 0" ± 5"
2. the vertical (down-hole)
semi-variogram: azimuth 90" ± 90", dip 90" ± 5"
For
the copper data, the average inter-borehole spacing is around 100 metres. For
the ‘between hole’ semi-variogram I have chosen an interval of 25 metres. For
the down-the-hole semi-variogram, I have chosen the length of the core
sections, 5 metres.
A
progress bar and variogram cloud plot will appear on your screen to let you
know that the calculation is proceeding.
With
our choices, all pairs of samples between 2.5 and 7.5 [5 ± 2.5] metres apart
will be grouped together in the vertical (down-hole) direction. For each of
these pairs, the difference in value will be calculated and squared. All of
these values will be added together and divided by twice the number of pairs.
This calculation will result in one point to be plotted on our final
semi-variogram graph. This process will be repeated for all pairs of samples
between 7.5 and 12.5 metres apart, and so on.
When
the calculation is finished you will be given the opportunity to see the final
graphs. New menus will appear at the top
of the screen:
When
you select 
from the 
menu, a new dialog will be displayed.
This
dialog lists all of the semi-variograms which have been calculated (and for
which the routine found pairs of samples).
You may plot any combination of these calculated semi-variograms on the
screen at once. Next to each calculated semi-variogram is a check box. At the
top of the dialog, you will see the message “You may select one or more of
these at one time”. Check the boxes for the ones you want to plot.
At
the right hand side of the dialog, you will see two small graphs. These
indicate the type of graph you can plot. There are two ways in which the graph
can be plotted. Firstly as a symbol for each calculated point on the graph. The
symbol size is proportional to the number of pairs of samples which have been
averaged to obtain that point.
Choosing
the two directions and clicking on the upper of the plotting options – the
scaled symbols plot – results in the following graph.
The
symbols in this graph are scaled to illustrate the number of pairs of samples
which were found in that interval. The largest symbol in this graph has 5545pairs
grouped together into one interval. The smallest point has only 9 pairs in its
calculation and is, one would think, somewhat less reliable.
If
we had chosen the shaded graph option, we would get the following graph.
This
display is produced as follows:
q For each calculated
semi-variogram, we join the first point to the third, the third to the fifth
and so on.
q The second point is joined
to the fourth, the fourth to the sixth and so on.
q The area between these two
lines is shaded.
This
display may be easier to interpret, especially for beginners. It does not,
however, give any information about the number of pairs of samples in each
interval. One single pair of samples giving an erratic
When
you have looked at the graphs to your heart’s content, you can choose to fit a
model to the calculated (or experimental)
semi-variogram.
Note
in particular the third option on the semi-variogram menu, which enables you to
store the experimental
semi-variograms - not the model - on a text file for input to (say) a report
quality graphics package. An EcoSSe option (read in experimental
semi-variograms) exists, which allows you to read this file back in and
continue with the modelling stage.
We
see that the two semi-variograms look very different in shape. They appear to
have similar nugget effects and final sills, but widely different ranges of
influence. We can, therefore, assume
that we have obvious geometric anisotropy.
In
point of fact, we constructed the semi-variograms in these particular
directions because we had a good prior idea that the dump is stratified
horizontally. If you have no preconceptions on which directions anisotropy
might take (if any), then the default directions are as good a place to start
as any.
Fitting a semi-variogram
model
It
would now be appropriate to fit a model to the experimental graph, so that we
can proceed to estimating unsampled locations. We will need to fit
semi-variograms in our major directions of anisotropy. Consider firstly the ‘horizontal’
semi-variogram:
This
looks pretty Spherical, although the nugget effect is difficult to pin down,
since we have few samples at the shorter distances on the graph.
On
the other hand, we have enormous numbers of pairs of samples in the longer
distance points, enabling us to get a pretty good handle on the range of
influence and sill for the semi-variogram model.
Using
the ‘point and click’ modelling facilities, described earlier in Tutorial Three
(Part 1) and Tutorial Four, we came up with the following model for the ‘all
horizontal’ semi-variogram:
Turning
to the vertical model, we bear in mind the sill and nugget effect found in this
direction and use that to guide our choices for the shorter range direction:
As
a measure of the goodness of fit of the model to the calculated points, the
Cressie goodness of fit statistic is quoted at the bottom of the dialog. We use
a modified version of this statistic which is standardised by the total number
of pairs included in the graph. This gives a figure which is not influenced by
the number of pairs and can, perhaps, be more objectively interpreted. The
statistic is calculated as follows:
It
is a good idea to get this as low as possible, but not at the expense of a good
visual fit.
Given
that the above model is an attractive fit to the points in the vertical
direction, we return to the horizontal direction and adjust the model to find
the longer range of influence.
By
the way, you cannot get this plot with the software. This was produced by
capturing the screens, pasting into MSPaintTM
and editing for clarity!
In
the EcoSSe
software, only geometric anisotropy is allowed. That is, you have a ‘standard’
semi-variogram in one direction. The models for the two other directions
orthogonal to this must have the same shape of semi-variogram model but can
differ in range of influence. When you begin kriging, you will specify the
‘anisotropy factors’ for the change in range of influence with direction.
The
important thing to note here is that the horizontal direction has to have a
range 14 times that of the vertical direction, if we are to keep the nugget
effect and sills the same in all directions.
If
you store a model on file, at this stage it will not contain anisotropy
information. Remember to update your model once you enter these – possibly
after cross validation.
Before
quitting this routine, we might want to store the calculated semi-variograms in
case we want to look at them again later.
You can store any combination of the calculated semi-variograms on a
file. EcoSSe will prompt you for the
name of the file. The default name is the original data file name with an
extension of .SXP. You can change the default
extension simply by typing in a new one. Alternatively you can change the whole
name to something entirely different.
Choosing
which semi-variograms to store is identical to choosing which ones to
plot:
The
stored semi-variograms can be read back in at any time using the option on the
main menu:
Storing a semi-variogram
model
You
may also want to store your model on a file for future use in modelling or in
the kriging routines:
You
will be prompted for the name of the output file on which the model will be
stored. This has a default name the same as your original data file and an
extension of .par.
The
model file is a flat text file listing all the possible semi-variogram
parameters and can be accessed by Wordpad, Notepad or some such for reporting
or editing.
Having
done all we need to do with the semi-variogram calculation and modelling:
which
returns you to the main menu bar.
This
Tutorial is continued in Tutorial Session 7 – 3D Ordinary Kriging.